Function – Complete Chapter Notes for JEE Main, N NIMCET, CUET
Table of Contents
- Definition of Function
- Representation of Functions
- Types of Functions
- Domain, Co-domain & Range
- Composition of Functions
- Inverse of a Function
- Even and Odd Functions
- Periodic Functions
- Algebra of Functions
- Important Standard Functions
- Graph Transformations
- Important Examples
- Summary Points (Exam-Oriented)
1. Definition of Function
A function is a rule that assigns every element of a non-empty set domain to exactly one element of a non-empty set co-domain.
We write the function as $f : A \to B$.
Example: If $f(x) = x^2$ and $A = \{1,2,3\}$, then range $= \{1,4,9\}$.
2. Representation of Functions
- Arrow Diagram: mapping arrows from domain to co-domain.
- Rule Method: e.g. $f(x) = 2x + 1$.
- Graph: set of all points $(x, f(x))$.
Vertical Line Test: A vertical line should intersect graph at most once.
3. Types of Functions
3.1 One-One (Injective)
$f(x_1) = f(x_2) \Rightarrow x_1 = x_2$
3.2 Many-One
Multiple inputs give same output.
3.3 Onto (Surjective)
Range = Co-domain.
3.4 Bijective
One-one and onto → Inverse exists.
3.5 Constant Function
$f(x) = c$
3.6 Identity Function
$I(x) = x$
3.7 Modulus Function
$f(x) = |x|$
3.8 Greatest Integer Function
$f(x) = \lfloor x \rfloor$
3.9 Fractional Part Function
$\{x\} = x - \lfloor x \rfloor$
3.10 Signum Function
$\text{sgn}(x) = 1, 0, -1$ based on sign of $x$
4. Domain, Co-domain & Range
4.1 Domain
- Denominator $\ne 0$
- Square root → expression $\ge 0$
- Log → argument $> 0$
- Inverse trigonometric → restricted inputs
Example (Domain):
For $f(x)=\frac{1}{\sqrt{5-2x}}$, condition is:
$5-2x > 0 \Rightarrow x < \frac{5}{2}$
Domain: $(-\infty, \frac{5}{2})$
4.2 Range
Actual output values of $f(x)$.
Example (Range):
$f(x)=\dfrac{1}{x^2+1}$
$x^2 \ge 0 \Rightarrow x^2+1 \ge 1$
Range → $(0,1]$
5. Composition of Functions
If $f : A\to B$ and $g : B\to C$, then $(g\circ f)(x)=g(f(x))$.
Example: If $f(x)=x+2$ and $g(x)=x^2$, then $(g\circ f)(x)=(x+2)^2$.
6. Inverse of a Function
Inverse exists only for bijective functions.
Steps:
- Write $y=f(x)$
- Swap $x,y$
- Solve for $y$ → this is $f^{-1}(x)$
Example: $f(x)=\frac{2x-1}{3}$
$f^{-1}(x)=\frac{3x+1}{2}$
7. Even and Odd Functions
Even Function:
$f(-x)=f(x)$
Odd Function:
$f(-x)=-f(x)$
Example: $x^2$ even, $x^3$ odd
8. Periodic Functions
A function $f$ is periodic if $f(x+T)=f(x)$.
- $\sin x$, $\cos x$ → $2\pi$
- $\tan x$ → $\pi$
Shortcut: For $f(x)=\sin(ax+b)$ → Period $=\frac{2\pi}{|a|}$
9. Algebra of Functions
- $(f+g)(x)=f(x)+g(x)$
- $(f-g)(x)=f(x)-g(x)$
- $(fg)(x)=f(x)g(x)$
- $(kf)(x)=k\cdot f(x)$
- $\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}$, $g(x)\ne 0$
10. Important Standard Functions
- Polynomial: $f(x)=ax^n+\cdots$
- Rational: $f(x)=\frac{P(x)}{Q(x)}$
- Exponential: $a^x$
- Logarithmic: $\log_a x$
- Trigonometric, inverse trigonometric
- Modulus, step, signum
11. Graph Transformations
- Vertical shift: $f(x)+k$
- Horizontal shift: $f(x-a)$
- Reflection in x-axis: $-f(x)$
- Reflection in y-axis: $f(-x)$
- Vertical stretch: $k f(x)$
- Horizontal stretch: $f(kx)$
12. Important Examples
Example 1: One-One / Many-One
$f(x)=x^2$ is many-one on $\mathbb{R}$ but one-one on $[0,\infty)$.
Example 2: Range
$f(x)=\sqrt{9-x^2}$ → Range $=[0,3]$
Example 3: Composition
For $f(x)=x^2+1$, $g(x)=\sqrt{x-1}$:
$(g\circ f)(x)=|x|$
Example 4: Inverse
$f(x)=\frac{2x-1}{3}$ → $f^{-1}(x)=\frac{3x+1}{2}$
13. Summary Points
- If $f$ is increasing → it is injective.
- Only bijective → inverse exists.
- For log → argument $>0$.
- For mod function → use piecewise.
- For range → convert to standard forms.
- Graphs of G.I.F., modulus, signum → must memorize.